Self-Adjointness, Symmetries, and Conservation Laws for a Class of ...

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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 407908, 6 pages http://dx.doi.org/10.1155/2013/407908

Research Article Self-Adjointness, Symmetries, and Conservation Laws for a Class of Wave Equations Incorporating Dissipation Yang Wang and Long Wei Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Zhejiang 310018, China Correspondence should be addressed to Long Wei; [email protected] Received 17 February 2013; Revised 30 April 2013; Accepted 13 May 2013 Academic Editor: Qian Guo Copyright Β© 2013 Y. Wang and L. Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we study the nonlinear self-adjointness and conservation laws for a class of wave equations with a dissipative source. We show that the equations are nonlinear self-adjoint. As a result, from the general theorem on conservation laws proved by Ibragimov and the symmetry generators, we find some conservation laws for this kind of equations.

1. Introduction There have been abundant literatures that have contributed to the studies of Lie symmetry classification of various classes of (1 + 1)-dimensional nonlinear wave equations and their individual members. Probably, Barone et al. [1] were the first to study the nonlinear wave equation 𝑒𝑑𝑑 = 𝑒π‘₯π‘₯ + 𝐹(𝑒) by means of symmetry method. Motivated by a number of physical problems, Ames et al. [2] investigated group properties of quasilinear 𝑒𝑑𝑑 = [𝑓(𝑒)𝑒π‘₯ ]π‘₯ . Later, Torrisi and Valenti [3, 4], generalizing above equations, have investigated the symmetries of the following equation: 𝑒𝑑𝑑 = [𝑓 (𝑒) 𝑒π‘₯ + 𝑔 (π‘₯, 𝑒)]π‘₯ .

(1)

Furthermore, classification results for the equation 𝑒𝑑𝑑 + 𝐾(𝑒)𝑒𝑑 = [𝐹(𝑒)𝑒π‘₯ ]π‘₯ can be found in [5, 6]. An expanded form of the latter equation 𝑒𝑑𝑑 + 𝐾 (𝑒) 𝑒𝑑 = [𝐹(𝑒)𝑒π‘₯ ]π‘₯ + 𝐻 (𝑒) 𝑒π‘₯

(2)

was studied by Kingston and Sophocleous [7]. In the papers mentioned above, many interesting results including Lie point and nonlocal symmetries classification were systematically investigated. In this paper, we consider a subclass of (2) with 𝐾(𝑒) = 𝛼: 𝑒𝑑𝑑 + 𝛼𝑒𝑑 βˆ’ [𝑓 (𝑒) 𝑒π‘₯ + 𝑔 (𝑒)]π‘₯ = 0,

functions, 𝑑 is the time coordinate, π‘₯ is the one-space coordinate, and 𝛼 is a nonzero constant. Here, we will focus on the nonlinear self-adjointness and conservation laws for (3). For (3), we cannot find easily the variational structure so it is inconvenient to apply the Noether theorem to construct conservation laws straightforward for this equation. However, it is fortunate that Ibragimov recently proved a result on conservation laws [8], which does not require the existence of a Lagrangian. The Ibragimov theorem on conservation laws provides an elegant way to establish local conservation laws for the equations under consideration. Since the seminal work of Ibragimov [8], more and more works are dedicated to studying the self-adjointness and conservation laws for some equations in mathematical physics and there are many new developments, including strict selfadjointness [9–11], quasi-self-adjointness [12–15], weak selfadjointness [16, 17], and nonlinear self-adjointness [18–21] and some results which have been communicated in the recent literature; for more references, see [21] and references therein. For the sake of completeness, we briefly present the notations, definition of nonlinear self-adjointness, and Ibragimov’s theorem on conservation laws. Let π‘₯ = (π‘₯1 , π‘₯2 , . . . , π‘₯𝑛 ) be 𝑛 independent variables, 𝑒 = 𝑒(π‘₯) a dependent variable,

(3)

which is also viewed as a special case of (1) with an additional dissipation, where 𝑓(𝑒) and 𝑔(𝑒) are arbitrary differentiable

𝑋 = πœ‰π‘–

πœ• πœ• +πœ‚ πœ•π‘₯𝑖 πœ•π‘’

(4)

2

Abstract and Applied Analysis

a symmetry of an equation 𝐹 (π‘₯, 𝑒(1) , 𝑒(2) , . . . , 𝑒(𝑠) ) = 0, πΉβˆ— (π‘₯, 𝑒(1) , V(1) , . . . , 𝑒(𝑠) , V(𝑠) ) :=

𝛿L =0 𝛿𝑒

(5) (6)

the adjoint equation to (5), where L = V𝐹 is called formal Lagrangian, V = V(π‘₯) is a new dependent variable, and 𝑠 πœ• πœ• 𝛿 = + βˆ‘ (βˆ’1)π‘š 𝐷𝑖1 β‹… β‹… β‹… π·π‘–π‘š 𝛿𝑒 πœ•π‘’ π‘š=1 πœ•π‘’π‘–1 β‹…β‹…β‹…π‘–π‘š

(7)

Definition 1. Equation (5) is said to be nonlinearly self-adjoint if the equation obtained from the adjoint equation (6) by the substitution V = πœ™(π‘₯, 𝑒) with a certain function πœ™(π‘₯, 𝑒) =ΜΈ 0 is identical with the original equation (5); in other words, the following equations hold: (8)

for some differential function πœ† = πœ†(π‘₯, 𝑒, 𝑒(1) , . . .). Particularly, if (8) holds for a certain function πœ™ such that πœ™π‘’ =ΜΈ 0 and πœ™π‘₯𝑖 =ΜΈ 0 for some π‘₯𝑖 , (5) is called weak self-adjoint; if (8) holds for a certain function πœ™ such that πœ™ = πœ™(𝑒) =ΜΈ 𝑒 and πœ™σΈ€  (𝑒) =ΜΈ 0, then (5) is called quasi-self-adjoint; if (8) holds for πœ™ = 𝑒, then (5) is called (strictly) self-adjoint. In the following, we recall the β€œnew conservation theorem” given by Ibragimov in [8]. We will find conservation laws for (3) by this theorem. Theorem 2. Any Lie point, Lie-BΒ¨acklund, and nonlocal symmetry 𝑋 = πœ‰π‘–

πœ• πœ• +πœ‚ πœ•π‘₯𝑖 πœ•π‘’

πœ•L πœ•L πœ•L βˆ’ 𝐷𝑗 ( ) + π·π‘˜ 𝐷𝑗 ( ) πœ•π‘’π‘– πœ•π‘’π‘–π‘— πœ•π‘’π‘–π‘—π‘˜ βˆ’π·π‘™ π·π‘˜ 𝐷𝑗 (

πœ•L ) + β‹…β‹…β‹…] πœ•π‘’π‘–π‘—π‘˜π‘™

πœ•L πœ•L + 𝐷𝑗 (π‘Š) [ βˆ’ π·π‘˜ ( ) + β‹…β‹…β‹…] πœ•π‘’π‘–π‘— πœ•π‘’π‘–π‘—π‘˜ + π·π‘˜ 𝐷𝑗 (π‘Š) [

In this section, we determine the nonlinearly self-adjoint subclasses of (3). Let 𝐹 = 𝑒𝑑𝑑 + 𝛼𝑒𝑑 βˆ’ 𝑓𝑒 (𝑒) 𝑒π‘₯2 βˆ’ 𝑓 (𝑒) 𝑒π‘₯π‘₯ βˆ’ 𝑔𝑒 (𝑒) 𝑒π‘₯ ; L = V (𝑒𝑑𝑑 + 𝛼𝑒𝑑 βˆ’ 𝑓𝑒 𝑒π‘₯2 βˆ’ 𝑓𝑒π‘₯π‘₯ βˆ’ 𝑔𝑒 𝑒π‘₯ ) .

(11)

(12)

Computing the variational derivative of this formal Lagrangian 𝛿L πœ•L πœ•L πœ•L πœ•L πœ•L βˆ’ 𝐷π‘₯ + 𝐷𝑑2 + 𝐷π‘₯2 , (13) = βˆ’ 𝐷𝑑 𝛿𝑒 πœ•π‘’ πœ•π‘’π‘‘ πœ•π‘’π‘₯ πœ•π‘’π‘‘π‘‘ πœ•π‘’π‘₯π‘₯ we obtain the adjoint equation of (3): πΉβˆ— = βˆ’π›ΌV𝑑 + V𝑑𝑑 βˆ’ 𝑓Vπ‘₯π‘₯ + 𝑔𝑒 Vπ‘₯ = 0.

(14)

Assume that πΉβˆ— |V=β„Ž(𝑑,π‘₯,𝑒) = πœ†πΉ, for a certain function πœ†, where 𝐹 is given by (11); then we have β„Žπ‘’ 𝑒𝑑𝑑 + β„Žπ‘’π‘’ 𝑒𝑑2 + (2β„Žπ‘‘π‘’ βˆ’ π›Όβ„Žπ‘’ ) 𝑒𝑑 βˆ’ π‘“β„Žπ‘’ 𝑒π‘₯π‘₯ βˆ’ π‘“β„Žπ‘’π‘’ 𝑒π‘₯2 + (𝑔𝑒 β„Žπ‘’ βˆ’ 2π‘“β„Žπ‘₯𝑒 ) 𝑒π‘₯ + β„Žπ‘‘π‘‘ βˆ’ 𝛼𝑒𝑑 βˆ’ π‘“β„Žπ‘₯π‘₯ + 𝑔𝑒 β„Žπ‘₯ (15) = πœ† (𝑒𝑑𝑑 + 𝛼𝑒𝑑 βˆ’ 𝑓𝑒 𝑒π‘₯2 βˆ’ 𝑓𝑒π‘₯π‘₯ βˆ’ 𝑔𝑒 𝑒π‘₯ ) . The comparison of the coefficients of 𝑒𝑑𝑑 , 𝑒𝑑2 , 𝑒π‘₯2 , 𝑒𝑑 , and 𝑒π‘₯ in both sides of (15) yields πœ† = β„Žπ‘’ ,

(16)

β„Žπ‘’π‘’ = 0,

(17)

πœ†π‘“π‘’ = π‘“β„Žπ‘’π‘’ ,

(18)

πœ†π›Ό = 2β„Žπ‘‘π‘’ βˆ’ π›Όβ„Žπ‘’ ,

(19)

πœ†π‘”π‘’ = 2π‘“β„Žπ‘₯𝑒 + 𝑔𝑒 β„Žπ‘’ ;

(20)

(9)

of (5) provides a conservation law 𝐷𝑖 (𝐢𝑖 ) = 0 for the system comprising (5) and its adjoint equation (6). The conserved vector is given by 𝐢𝑖 = πœ‰π‘– L + π‘Š [

2. Nonlinear Self-Adjointness of (3)

then we have the following formal Lagrangian for (3):

denotes the Euler-Lagrange operator. Now, let us state the definition of nonlinear self-adjointness for a equation.

πΉβˆ— |V=πœ™ = πœ† (π‘₯, 𝑒, 𝑒(1) , . . .) 𝐹

The paper is organized as the follows. In Section 2, we discuss the nonlinear self-adjointness of (3). In Section 3, we establish conservation laws for some particular cases of (3) using Theorem 2.

thus, (15) is reduced as β„Žπ‘‘π‘‘ βˆ’ π›Όβ„Žπ‘‘ βˆ’ π‘“β„Žπ‘₯π‘₯ + 𝑔𝑒 β„Žπ‘₯ = 0.

(21)

It follows from (17) and (18) that πœ†π‘“π‘’ = 0.

(10)

πœ•L πœ•L βˆ’ 𝐷𝑙 ( ) + β‹…β‹…β‹…] + β‹…β‹…β‹… , πœ•π‘’π‘–π‘—π‘˜ πœ•π‘’π‘–π‘—π‘˜π‘™

where π‘Š = πœ‚ βˆ’ πœ‰π‘— 𝑒𝑗 is the Lie characteristic function and L = V𝐹 is the formal Lagrangian.

(22)

Equation (22) splits into the following two cases. Case 1 (πœ† = 0). In this case, from (16) we get that β„Žπ‘’ = 0; thus, (16)–(20) are all satisfied. Noting that β„Ž does not depend on the function 𝑒, 𝑓 and 𝑔 are two functions of 𝑒, so from (21) we obtain that β„Žπ‘‘π‘‘ βˆ’ π›Όβ„Žπ‘‘ = 0,

βˆ’π‘“β„Žπ‘₯π‘₯ + 𝑔𝑒 β„Žπ‘₯ = 0.

(23)

Abstract and Applied Analysis

3 Table 2: Symmetries of (3) for some special choices of 𝑓 and 𝑔.

Table 1: Nonlinear self-adjointness of (3). 𝑓 βˆ€(𝑓𝑒 =ΜΈ 0) βˆ€(𝑓𝑒 =ΜΈ 0)

𝑔

V

βˆ€ 𝐢5

βˆ€(𝑓𝑒 =ΜΈ 0) 𝐢5 ∫ 𝑓(𝑒)𝑑𝑒 𝐢0 ( =ΜΈ 0)

𝐢1 + 𝐢2 𝑒

𝐢1 + 𝐢2 𝑒𝛼𝑑 (𝐢1 + 𝐢2 𝑒𝛼𝑑 )(𝐢3 + 𝐢4 π‘₯) 𝐢1 + 𝐢2 𝑒𝛼𝑑 +(𝐢3 + 𝐢4 𝑒𝛼𝑑 )𝑒π‘₯/𝐢5 𝐢1 𝑒(𝐢2 /𝐢0 )π‘₯+𝛼𝑑 𝑒 + 𝑏(𝑑, π‘₯)

Selfadjointness Nonlinear Nonlinear

𝑔

𝑓 𝑒

βˆ’2

Symmetries πœ• πœ• 𝑋3 = π‘₯ βˆ’π‘’ πœ•π‘₯ πœ•π‘’ πœ• 2𝑒 πœ• 𝑋3 = π‘₯ + πœ•π‘₯ 𝑛 πœ•π‘’ πœ• 𝑋3 = 𝑒 , πœ•π‘’ πœ• πœ• 1 πœ• 𝑋4 = 𝑑 + π‘₯ βˆ’ (𝑑 + 𝛼π‘₯)𝑒 πœ•π‘₯ πœ•π‘‘ 2 πœ•π‘’ πœ• πœ• πœ• 1 πœ• 𝑋3 = 𝑒 , 𝑋4 = 𝑑 + π‘₯ βˆ’ 𝛼π‘₯𝑒 πœ•π‘’ πœ•π‘₯ πœ•π‘‘ 2 πœ•π‘’

ln 𝑒

𝑒𝑛 (𝑛 =ΜΈ 0)

1

1

𝑒

1

1

Nonlinear Weak

𝛼 is a nonzero constant; 𝑏(𝑑, π‘₯) satisfies (27).

Equation (23) can be satisfied by taking some appropriate function β„Ž = β„Ž(𝑑, π‘₯). Therefore, we can take V = β„Ž(π‘₯, 𝑑) =ΜΈ 0 such that πΉβˆ— |V=β„Ž(𝑑,π‘₯,𝑒) = πœ†πΉ holds. Thus, (3) is nonlinear selfadjoint in this case. Case 2 (𝑓𝑒 = 0). That is, 𝑓 is a constant with respect to 𝑒; without loss of generality, we set 𝑓 = 𝐢0 =ΜΈ 0. From (17), we assume that β„Ž (𝑑, π‘₯, 𝑒) = π‘Ž (𝑑, π‘₯) 𝑒 + 𝑏 (𝑑, π‘₯) .

𝐢0 π‘˜σΈ€  (π‘₯) , π‘˜ (π‘₯)

(25)

which implies that 𝑔(𝑒) = 𝐢2 𝑒 + 𝐢1 is a linear function of 𝑒 and π‘˜(π‘₯) = 𝐢1 𝑒(𝐢2 /𝐢0 )π‘₯ , where 𝐢2 and 𝐢1 are constants. Hence, we have β„Ž (𝑑, π‘₯, 𝑒) = 𝐢1 𝑒(𝐢2 /𝐢0 )π‘₯+𝛼𝑑 𝑒 + 𝑏 (𝑑, π‘₯) .

𝐷𝑑 (𝐢𝑑 ) + 𝐷π‘₯ (𝐢π‘₯ ) = 0.

3.1. Lie Symmetries. Now, let us show the Lie classical symmetries for (3). Based on Lie group theory [22], we assume that a Lie point symmetry of (3) is a vector field

𝑋(2) = πœ‰

3. Symmetries and Conservation Laws In this section, we will apply Theorem 2 to construct some conservation laws for (3). First, we show the Lie classical

πœ• πœ• πœ• πœ• +πœ™ +πœ‚ + πœ‚π‘₯(1) πœ•π‘₯ πœ•π‘‘ πœ•π‘’ πœ•π‘’π‘₯

+

(27)

Theorem 3. Let 𝛼 be a nonzero constant, 𝑓(𝑒) an arbitrary function of 𝑒, and the adjoint equation of (3) given by (14); then (3) is nonlinear self-adjointness with the substitution V and function 𝑔(𝑒) given in Table 1.

(29)

on R+ Γ— R Γ— R such that 𝑋(2) 𝐹 = 0 when 𝐹 = 0, and taking into account (3), the operator 𝑋(2) is given as follows:

(26)

which is easy to be solved. In this case, 𝑓(𝑒) = 𝐢0 is a constant, 𝑔(𝑒) = 𝐢2 𝑒 + 𝐢1 , and πœ† = β„Žπ‘’ = 𝐢1 𝑒(𝐢2 /𝐢0 )π‘₯+𝛼𝑑 ; therefore, (3) is a weak self-adjoint. Here, we omit the tedious calculations to obtain the solutions of (23) and (27). In Table 1, we summarize the classification of nonlinear self-adjointness of (3) with the conditions that 𝑓(𝑒) and 𝑔(𝑒) should satisfy. In what follows, the symbol for all means that the corresponding function has no restrictions and 𝑐𝑖 (1 ≀ 𝑖 ≀ 5) are arbitrary constants. Thus, we have demonstrated the following statement.

πœ• πœ• πœ• + πœ™ (𝑑, π‘₯, 𝑒) + πœ‚ (𝑑, π‘₯, 𝑒) πœ•π‘₯ πœ•π‘‘ πœ•π‘’

𝑋 = πœ‰ (𝑑, π‘₯, 𝑒)

Substituting (26) into (22) and using the original equation (3), we derive that 𝑏(𝑑, π‘₯) satisfies 𝑏𝑑𝑑 βˆ’ 𝛼𝑏𝑑 βˆ’ 𝐢0 𝑏π‘₯π‘₯ + 𝐢2 𝑏π‘₯ = 0,

(28)

(24)

Taking into account (16) and (19), we have π‘Žπ‘‘ (𝑑, π‘₯) = π›Όπ‘Ž(𝑑, π‘₯); thus, π‘Ž(𝑑, π‘₯) = π‘˜(π‘₯)𝑒𝛼𝑑 . From (20), we deduce that 𝑔𝑒 (𝑒) =

symmetries for (3) for some special choice of 𝑓 and 𝑔. Then applying formula in Theorem 2 to the formal Lagrangian (12), and to the symmetries 𝑋𝑖 and eliminating V by the substitution V given in Table 1, we obtain the conservation law

πœ‚π‘‘(1)

πœ• πœ• (2) πœ• + πœ‚π‘₯π‘₯ + πœ‚π‘‘π‘‘(2) , πœ•π‘’π‘‘ πœ•π‘’π‘₯π‘₯ πœ•π‘’π‘‘π‘‘

(30)

where πœ‚π‘₯(1) = 𝐷π‘₯ πœ‚ βˆ’ (𝐷π‘₯ πœ‰) 𝑒π‘₯ βˆ’ (𝐷π‘₯ πœ™) 𝑒𝑑 , πœ‚π‘‘(1) = 𝐷𝑑 πœ‚ βˆ’ (𝐷𝑑 πœ‰) 𝑒π‘₯ βˆ’ (𝐷𝑑 πœ™) 𝑒𝑑 , (2) = 𝐷π‘₯ (πœ‚π‘₯(1) ) βˆ’ (𝐷π‘₯ πœ‰) 𝑒π‘₯π‘₯ βˆ’ (𝐷π‘₯ πœ™) 𝑒π‘₯𝑑 , πœ‚π‘₯π‘₯

(31)

πœ‚π‘‘π‘‘(2) = 𝐷𝑑 (πœ‚π‘‘(1) ) βˆ’ (𝐷𝑑 πœ‰) 𝑒π‘₯𝑑 βˆ’ (𝐷𝑑 πœ™) 𝑒𝑑𝑑 . The condition 𝑋(2) 𝐹 = 0, when 𝐹 = 0, will yield determining equations. Solving these determining equations, we can obtain the symmetries of (3). For 𝑓(𝑒) and 𝑔(𝑒) arbitrary, the symmetries that are admitted by (3) are 𝑋1 =

πœ• , πœ•π‘₯

𝑋2 =

πœ• . πœ•π‘‘

(32)

For some special choices of the functions 𝑓(𝑒) and 𝑔(𝑒), here we omit the details of routine calculations and present the symmetries (besides 𝑋1 and 𝑋2 ) that are admitted by (3) in Table 2.

4

Abstract and Applied Analysis Table 3: Conservation laws of (3) for some special choices of 𝑓 and 𝑔 (𝑛 =ΜΈ 0).

𝑓

𝑔

1 𝑒2

ln 𝑒

𝑒𝑛

Conserved vector with substitution V 𝐢𝑑 = (V𝑑 βˆ’ 𝛼V)(𝑒 + π‘₯𝑒π‘₯ ) βˆ’ V(𝑒𝑑 + π‘₯𝑒π‘₯𝑑 )

Symmetries π‘₯

π‘₯

1

πœ• πœ• βˆ’π‘’ πœ•π‘₯ πœ•π‘’

𝐢π‘₯ = π‘₯V𝑒𝑑𝑑 + 𝛼π‘₯V𝑒𝑑 + V V = 𝐢1 + 𝐢2 𝑒𝛼𝑑 2 2 𝐢𝑑 = ( 𝑒 βˆ’ π‘₯𝑒π‘₯ ) (𝛼V βˆ’ V𝑑 ) + V ( 𝑒𝑑 βˆ’ π‘₯𝑒π‘₯𝑑 ) 𝑛 𝑛 2 2 𝐢π‘₯ = π‘₯ (V𝑒𝑑𝑑 + 𝛼V𝑒𝑑 βˆ’ 𝑒𝑛 𝑒π‘₯ Vπ‘₯ ) + 𝑒𝑛+1 Vπ‘₯ βˆ’ (1 + ) V𝑒𝑛 𝑒π‘₯ 𝑛 𝑛 V = (𝐢1 + 𝐢2 𝑒𝛼𝑑 ) (𝐢3 + 𝐢4 π‘₯)

πœ• 2𝑒 πœ• + πœ•π‘₯ 𝑛 πœ•π‘’

1 1 (𝑑 + 𝛼π‘₯) 𝑒𝑑 + (1 + 𝛼𝑑 + 𝛼2 π‘₯) 𝑒 2 2 1 + 𝑑𝑒π‘₯𝑑 + π‘₯𝑒π‘₯π‘₯ ] + V𝑑 [ (𝑑 + 𝛼π‘₯) 𝑒 + π‘₯𝑒𝑑 + 𝑑𝑒π‘₯ ] 2 1 1 = V [(1 + π‘₯ + 𝛼𝑑)𝑒𝑑 + (𝑑 + 𝛼π‘₯)𝑒π‘₯ + (𝑑 + 𝛼 + 𝛼π‘₯)𝑒 2 2 1 + 𝑑𝑒𝑑𝑑 + π‘₯𝑒π‘₯𝑑 ] βˆ’ Vπ‘₯ [ (𝑑 + 𝛼π‘₯) 𝑒 + π‘₯𝑒𝑑 + 𝑑𝑒π‘₯ ] 2

𝐢𝑑 = βˆ’V [(1 + π‘₯ + 𝛼𝑑) 𝑒π‘₯ + 1

𝑒

𝑑

πœ• πœ• (𝑑 + 𝛼π‘₯)𝑒 πœ• +π‘₯ βˆ’ πœ•π‘₯ πœ•π‘‘ 2 πœ•π‘’ 𝐢π‘₯

V = 𝐢1 + 𝐢2 𝑒𝛼𝑑 + (𝐢3 + 𝐢4 𝑒𝛼𝑑 )𝑒π‘₯ or 𝑏(𝑑, π‘₯)

1

1

𝑑

𝛼 𝛼2 𝐢𝑑 = βˆ’V ( π‘₯𝑒𝑑 + π‘₯𝑒π‘₯π‘₯ + π‘₯𝑒 + 𝛼𝑑𝑒π‘₯ + 𝑒π‘₯ + 𝑑𝑒π‘₯𝑑 ) 2 2 𝛼 +V𝑑 (π‘₯𝑒𝑑 + 𝑑𝑒π‘₯ + π‘₯𝑒) 2 𝛼 𝛼 π‘₯ 𝐢 = V ( π‘₯𝑒π‘₯ + 𝑑𝑒𝑑𝑑 + 𝑒 + 𝛼𝑑𝑒𝑑 + 𝑒𝑑 + π‘₯𝑒π‘₯𝑑 ) 2 2 𝛼 βˆ’Vπ‘₯ (π‘₯𝑒𝑑 + 𝑑𝑒π‘₯ + π‘₯𝑒) 2

πœ• πœ• 𝛼π‘₯𝑒 πœ• +π‘₯ βˆ’ πœ•π‘₯ πœ•π‘‘ 2 πœ•π‘’

V = (𝐢1 + 𝐢2 𝑒𝛼𝑑 )(𝐢3 + 𝐢4 π‘₯) or 𝐢1 𝑒𝛼𝑑 𝑒 + 𝑏(𝑑, π‘₯)

3.2. Conservation Laws. For the symmetries 𝑋, from the formula (10) in Theorem 2, we can obtain readily some conservation laws for (3). For example, we take 𝑓(𝑒) = π‘’βˆ’2 and 𝑔(𝑒) = ln 𝑒; let us construct the conserved vector corresponding to the time translation group with the generator 𝑋2 =

πœ• . πœ•π‘‘

(33)

For this operator, we have πœ‰π‘‘ = 1, πœ‰π‘₯ = 0, πœ‚ = 0, and π‘Š = βˆ’π‘’π‘‘ . In this case, (3) is nonlinear self-adjoint and becomes 𝐹 = 𝑒𝑑𝑑 + 𝛼𝑒𝑑 + 2π‘’βˆ’3 𝑒π‘₯2 βˆ’ π‘’βˆ’2 𝑒π‘₯π‘₯ βˆ’ π‘’βˆ’1 𝑒π‘₯ .

(34)

The formal Lagrangian is L = V (𝑒𝑑𝑑 + 𝛼𝑒𝑑 + 2π‘’βˆ’3 𝑒π‘₯2 βˆ’ π‘’βˆ’2 𝑒π‘₯π‘₯ βˆ’ π‘’βˆ’1 𝑒π‘₯ ) ,

(35)

and the adjoint equation of (34) is Lβˆ— = V𝑑𝑑 βˆ’ 𝛼V𝑑 βˆ’ π‘’βˆ’2 Vπ‘₯π‘₯ + π‘’βˆ’1 Vπ‘₯ .

Therefore, we obtain the following conserved vector: 𝐢𝑑 = L + π‘Š (

πœ•L πœ•L πœ•L βˆ’ 𝐷𝑑 ) + (𝐷𝑑 π‘Š) πœ•π‘’π‘‘ πœ•π‘’π‘‘π‘‘ πœ•π‘’π‘‘π‘‘

= V (𝑒𝑑𝑑 + 𝛼𝑒𝑑 + 2π‘’βˆ’3 𝑒π‘₯2 βˆ’ π‘’βˆ’2 𝑒π‘₯π‘₯ βˆ’ π‘’βˆ’1 𝑒π‘₯ ) βˆ’ 𝑒𝑑 (𝛼V βˆ’ V𝑑 ) βˆ’ 𝑒𝑑𝑑 V = 2V𝑒π‘₯2 π‘’βˆ’3 βˆ’ V𝑒π‘₯π‘₯ π‘’βˆ’2 βˆ’ V𝑒π‘₯ π‘’βˆ’1 + 𝑒𝑑 V𝑑 , 𝐢π‘₯ = π‘Š (

πœ•L πœ•L πœ•L βˆ’ 𝐷π‘₯ ) + (𝐷π‘₯ π‘Š) πœ•π‘’π‘₯ πœ•π‘’π‘₯π‘₯ πœ•π‘’π‘₯π‘₯

= π‘’βˆ’1 𝑒𝑑 (V βˆ’ 2π‘’βˆ’2 V𝑒π‘₯ βˆ’ π‘’βˆ’1 Vπ‘₯ ) + π‘’βˆ’2 V𝑒π‘₯𝑑 , where V satisfies (36). The reckoning shows that the vector (37) satisfies the conservation equation (28). In this case, since (3) is nonlinear self-adjoint, from Table 1 we take V = 𝐢1 + 𝐢2 𝑒𝛼𝑑 . So, the conserved vector is simplified as follows: 𝐢𝑑 = 2V𝑒π‘₯2 π‘’βˆ’3 βˆ’ V𝑒π‘₯π‘₯ π‘’βˆ’2 βˆ’ V𝑒π‘₯ π‘’βˆ’1 + 𝑒𝑑 V𝑑 ,

(36)

(37)

𝐢π‘₯ = π‘’βˆ’1 𝑒𝑑 V (1 βˆ’ 2π‘’βˆ’2 𝑒π‘₯ ) + π‘’βˆ’2 V𝑒π‘₯𝑑 .

(38)

Abstract and Applied Analysis

5

Particularly, if setting V = 1, we have 𝑑

𝐢 = βˆ’π‘’π‘₯ 𝑒

βˆ’1

βˆ’2

βˆ’ 𝐷π‘₯ (𝑒 𝑒π‘₯ ) ,

π‘₯

βˆ’1

βˆ’2

𝐢 = 𝑒 𝑒𝑑 + 𝐷𝑑 (𝑒 𝑒π‘₯ ) . (39)

Thus, the conserved vector can be reduced to the form 𝐢𝑑 = βˆ’π‘’π‘₯ π‘’βˆ’1 ,

𝐢π‘₯ = π‘’βˆ’1 𝑒𝑑 .

(40)

If setting V = 𝑒𝛼𝑑 , then the conserved vector (37) is simplified as 𝐢𝑑 = 𝛼𝑒𝛼𝑑 𝑒𝑑 βˆ’ 𝐷π‘₯ [𝑒𝛼𝑑 (π‘’βˆ’2 𝑒π‘₯ + ln 𝑒)] , 𝐢π‘₯ = βˆ’π›Όπ‘’π›Όπ‘‘ (π‘’βˆ’2 𝑒π‘₯ + ln 𝑒) + 𝐷𝑑 [𝑒𝛼𝑑 (π‘’βˆ’2 𝑒π‘₯ + ln 𝑒)] ;

(41)

therefore, it can be reduced as 𝐢𝑑 = 𝛼𝑒𝛼𝑑 𝑒𝑑 ,

𝐢π‘₯ = βˆ’π›Όπ‘’π›Όπ‘‘ (π‘’βˆ’2 𝑒π‘₯ + ln 𝑒) .

(42)

In what follows, we omit the tedious calculations and list only the conservation laws of (3) for some special choices of functions 𝑓(𝑒) and 𝑔(𝑒) in Table 3. In Table 3, the function 𝑏(𝑑, π‘₯) satisfies (27), V and the symmetry are taken from Tables 1 and 2, respectively. The reckoning shows that the vector listed in Table 3 satisfies the conservation equation (28) with corresponding substitution V. In the same way above, we can simplify the conserved vector using corresponding substitution V.

4. Conclusions Recently, the new outstanding concepts of nonlinear selfadjoint equations, containing quasi-self-adjoint and weak self-adjoint equations, which extend the self-adjointness to a more generalized meaning, have been introduced in order to find formal Lagrangians of differential equations without variational structure. Using these concepts and the general theorem on conservation laws that is, developed recently [8], nonlinear self-adjointness and conservation laws for (3) for different classes of 𝑓(𝑒) and 𝑔(𝑒) have been discussed. These conservation laws may be useful in mathematical analysis as they provide basic conserved quantity for obtaining various estimates for smooth solutions and defining suitable norms for weak solutions. Furthermore, it could make the construction of the bi-Hamiltonian form easier.

Acknowledgments The authors would like to express their sincere gratitude to the referees for many valuable comments and suggestions which served to improve the paper. The first author is partly supported by NSFC under Grant no. 11101111. The second author is partly supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12A01003 and subjects research and development foundation of Hangzhou Dianzi University under Grant no. ZX100204004-6.

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